The wave function for the Is state of an electron in the hydrogen atom is VIs(P) = e-p/ao where ao is the Bohr radius. The probability of finding the electron in a region W of R³ is equal to J, P(x, y, 2) dV where, in spherical coordinates, p(p) = |V1s(P)² Use integration in spherical coordinates to show that the probability of finding the electron at a distance greater than the Bohr radius is equal to 5/e = 0.677. (The Bohr radius is ao =5.3 x 10-1" m, but this value is not needed.)
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- For a hydrogen-like atom with atomic number of Z, calculate the expectation values of (a) r and (b) potential energy (U = - Zee²) in Απερτ, the 2s state. Note that the wave function for the 2s state is 1 √3²2π (2) ³/2 (2. )e-(z/2a0)r. 425 = Z aoDeduce the Uncertainty Principle Axap = 15/2/ from ΔΕΔΕΣ Τ = ħ/2Find the directions in space where the angular probability density for the l = 2, ml = 0 electron in hydrogen has its maxima and minima.
- The ground-state wave function for the electron in a hydro- gen atom is 1 1,(7) = VTa where r is the radial coordinate of the electron and a, is the Bohr radius. (a) Show that the wave function as given is normalized. (b) Find the probability of locating the electron between r, = a,/2 and r, = 3a,/2.Look up the values of the quantities in aB = h2 / 4π2 me kqe2 ,and verify that the Bohr radius aB is 0.529 x 10-10 m .What do we need to do to average over Θ and ф to get the probability that the electron is inside a shell of radius r and thickness dr?Problem 1. Consider the following 1-dimensional model of negative ion photodetachment. Let m be the electron mass, and assume the electron-atom potential energy is an attractive delta function, V(x)= - V. 8(x) with coefficient -Vo. (b) Now suppose a weak time-dependent perturbing electric field is applied along the x-axis, equal to E(t)= E, exp(-i w t) +c.c.; Find the energy-normalized final state eigenfunctions for the unperturbed Hamiltonian, having odd parity. (c) Using the Golden Rule, calculate the photodetachment rate as a function of (hbar w) in these units, i.e. calculate the probability per unit time in these units for the electron to escape to infinity. Glve an analytic expression and plot it from 0 final state energy up to an energy|10 E(ground) |, i.e. for frequencies that reach up to a final state energy that is 10 times the absolute value of the ground state energy. For making a plot of these numerical values, use V. =0.03 and a field strength amplitude Eo=0.0002.